# Operations on Matrices

## •    Let A = [aij] m x n and B = [bij] m x n be two matrices of same order m x n. Then the sum of the two matrices A and B is defined as a matrix C = [cij] m x n  where cij = aij + bij   for all possible values of i and j.•    If A and B are matrices of same order, then A + B =  B + A. This is commutative property of matrices under addition. Addition of matrices also holds the associative law.•    For any matrix A, there exists a null matrix O of the same order such that A + O = A = O + A.•    For any matrix A, there exists a matrix  – A such that: •    A + (– A) = O = (–A) + A •    Two matrices are subtracted by subtracting corresponding  elements of these matrices. •    Let A = [aij] m x n is a matrix and k is a scalar, then kA is another matrix, which is obtained by multiplying each element of A by a scalar k.•    The product AB of two matrices A and B is defined if numbers of columns of A is equal to the number of rows of B.   •    Matrix multiplication is not commutative in general.•    Matrix multiplication is associative whenever both sides of equality are defined.•    Matrix multiplication is distributive over matrix addition.•    For every square matrix A, there is an identity matrix of same order such that IA = AI = A.•    The product of two matrices can be null matrix while neither of them is a null matrix. If A is an m × n matrix and O is a null matrix, then •    the product of matrix A with the null matrix O is always a null matrix.•    In case of matrix multiplication if AB = O, then it does not necessarily imply that BA = O.•    The transpose of a matrix is obtained by interchanging its rows and columns. It is denoted by  A' or  AT. •    Properties of Transpose:•    For any square matrix A with real number entries, A + AT is a symmetric matrix and A – AT is a skew symmetric matrix.•    Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix. Keywords: Addition of matrices, Multiplication of matrices, Difference of matrices, Transpose of a matrix, Multiplication of a matrix by a scalar, Symmetric matrix, Skew symmetric matrix

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• Q1

If A and B are two matrices such that A + B and AB are both defined, then

Marks:1

A and B are square matrices of same order.

##### Explanation:
In square matrices, both addition and multiplication are defined. Therefore, A and B are square matrices of same order.
• Q2

If A is a square matrix, then AAT is always a

Marks:1

symmetric matrix.

##### Explanation:

(AAT )T = (AT )T A= AAT

Hence AAT is a symmetric matrix.

• Q3

Marks:1

##### Explanation:

• Q4

If A and B are symmetric matrices of the same order, then (AB – BA) is

Marks:2

A skew-symmetric matrix

##### Explanation:

Let A and B be symmetric.

Then At = A and Bt = B

(AB – BA)t = (AB)t – (BA)t

= BtAt – AtBt = BA – AB = – (AB – AB)

Hence, (AB – BA) is skew-symmetric.

• Q5

If A is skew-symmetric, then A3 is :

Marks:1

skew-symmetric

##### Explanation:

Let A be skew symmetric

Then, At = – A

(A3)t = (AAA)t = At. At. At

= (–A) (–A) (–A) = – A3

Hence, A3 is skew-symmetric